Learn Russian: Собака здесь?

In Russian, you don't need to change the word order to ask a yes/no question. You just change your intonation! In speech, raise your pitch sharply on the word you are asking about (in this case, здесь). In writing, a question mark is all you need.

Where is the word for "is" in this sentence?

Russian drops the verb "to be" (am, is, are) in the present tense. You just put the subject (собака) and the location (здесь) right next to each other. Literally, it translates to "Dog here?".

How do I reply to this question using the words for "yes" or "no"?

You can just say Да (Yes) or Нет (No). If you want to be a bit more specific without repeating the whole sentence, you can simply reply Да, здесь (Yes, here) or Нет, не здесь (No, not here).

What does the soft sign (ь) do at the end of здесь?

The soft sign doesn't make a sound on its own, but it changes how the consonant right before it is pronounced. It makes the с "soft," meaning you raise the middle of your tongue toward the roof of your mouth when making the "s" sound.